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Local Approximation of PageRank and Reverse PageRank

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### Local Approximation of PageRank and Reverse PageRank

Li-Tal Mashiach

Advisor: Dr. Ziv Bar-Yossef

13/03/08

- Review of PageRank
- Local PageRank approximation
- Algorithm
- Lower bounds
- PageRank vs. Reverse PageRank
- Applications of Reverse PageRank

PageRank

Most search engines analyze the hyperlink structure to order search results

PageRank

- Important measure of ranking for all major search engines

Sum of the

in-neighbors’ ranks

Review of PageRankRank divided among

all out-neighbors

Damping factor

- A random surfer is visiting the web:
- With probability , selects a random out-link
- With probability jumps to a random web page

- Run power method
- Initialize:
- Repeat until convergence:

- Challenges:
- Holding the whole web graph
- Multiplying a matrix by a vector

Local PR Approximation

Global PR calculates PR to all pages

Sometime we are interested in the PR of a small number of pages

- Person interested in the PR of his homepage
- Online business is interested in the PR of his own website and his competitors’ website
Do we need to calculate the PR of the whole graph for that?

- Given: local access to a directed graph G and target node
- Output: PR(u)
- local access:
- Cost: Number of queries to the link server

Link Server

Problem Statement[Chen, Gan, Suel, 2004]Overview

Review of PageRank

Local PageRank approximation

Algorithm

Lower bounds

PageRank vs. Reverse PageRank

Applications of Reverse PageRank

- inft(v,u) – the fraction of the PR score of v that flows to u on paths of length t

v

u2

u1

Another Characterization of PR[Jeh, Widom, 2003]u

- PRr(u) – PR score that flows into u from nodes at distance at most r from u
Theorem:

v

u2

u1

Another Characterization of PR[Jeh, Widom, 2003]u

Local PR Brute Force Algorithm[Chen, Gan, Suel, 2004]

- Goal: calculate PRr(u) for a sufficiently large r
- Algorithm:
- Crawl backwards the sub-graph of radius r around u
- For each node v at layer t calculate the inft(v,u)
- Sum up the weighted influence values

v

w1

w2

u

Optimization by Pruning

Heuristic to improve the cost

Prune all nodes whose influence is below some threshold

Was shown empirically to be sometimes better [Chen, Gan, Suel, 2004]

u

Analysis of the Algorithm

- This algorithm requires at most queries
- r – number of iterations until the PR random walk almost converges
- d – maximum in-degree of the graph

- In case of slow PR convergence or high in-degree, the algorithm is not feasible

- In the web graph there are a lot of web pages with high in-degree
- Conclusion: The algorithm is frequently unsuitable for the web graph
- Is this a limitation of this
specific algorithm only?

Lower Bounds in-degree

- Local PR approx. is hard for graphs with:
- High in-degree nodes
- Slow convergence of the PR random walk

Proof in-degree

x1

x2

x3

xm

- By reduction from the OR problem

Input:

Output:

queries are needed even for

randomized algorithms

The Reduction in-degree

1

1

0

m

X=

Gx=

…

…

….

…

u

- A - Alg. that calculates local PR
- B - Alg. that computes the OR function

The Reduction in-degree

1

1

0

m

X=

Gx=

Claim 1: Let |x| be the

number of 1’s in x. Then,

…

…

….

…

u

Claim 2: When ,

Proof Cont. in-degree

- Given an input x, B simulates A on Gx, u
- If PRx(u) ≥ p1 => OR=1
- If PRx(u) ≤ p0 => OR=0
- It means that the maximum number of queries A uses ≥

Conclusion

PageRank vs. Reverse PageRank web graph

- The local approximation algorithm should perform better on the Reverse Web Graph

Experimental Setup web graph

280,000 page crawl of the www.stanford.edu domain

22,000 page crawl of the www.cnn.com site

Convergence Rate web graph

Crawl Growth Rate web graph

In-deg: 38,606Out-deg: 255

Performance of the Algorithm web graph

Applications of Reverse PageRank web graph

Local RPR app.

Novel app.

TrustRank

Influencers in social networks

Hub web pages

Measuring semantic relatedness

Finding crawl seeds

Influencers in web graphSocial Networks

Goal: Market a new product to be adopted by a large fraction of a social network

Method:

- Initially target a few influential members
- Trigger a word of mouth process
- Results in a large number of users
How should we choose these seed members?

- Nodes with high RPR web graph
- Have short paths to many other nodes in the network
- Frequently the only gateways to these nodes

Influencers in Social Networks web graph

Influencers in Social Networks web graph

4-level BFS crawl

1-level BFS crawl

www.Livejournal.com, 3.5 million nodes

Hub Web Pages web graph

Goal: Find good starting points for search

- Difficult to formulate queries
- Broad search tasks
- Need to understand the surrounding context
Method: Find pages with short paths to many relevant pages

Why RPR?[Fogaras, 2003]

Hub Web Pages web graph

Fraction of hubs in the top 20 results for the queries:

1. “computer scientists”

2. “global warming”

3. “folk dancing”

4. “queen Elizabeth”

Meta-search engine over Yahoo! search

Measuring Semantic Relatedness web graph

Goal: Find the relatedness between two concepts

- For Natural language processing applications
Method: Use a taxonomy like the ODP or Wikipedia

Why RPR? web graph

b is a strong sub-concept of a in a taxonomy if

- there are many short paths from a to b
RPR- measure of b as sub-concept of a

RPR Similarity- two concepts will be similar in case they have significant overlap between their RPR vectors

- similarity between the vectors RPRa and RPRb

Measuring Semantic Relatedness web graph

Relatedness to “Einstein”

Relatedness to “Computer”

Agriculture

Physics Prize

Newton Isaac

Internet

0.6

0.6

-0.4

www.dmoz.org taxonomy

WordSimilarity-353

Finding Crawl Seeds web graph

Goal: Discover quickly new content on the web while incurring as little overhead as possible

- Overhead: old pages / new pages
Method: Find good seeds

- A page web graphp has high RPR if
- Many pages are reachable from p by short paths
- These pages are not reachable from many other pages

u

Known page

Why RPR?v

Unknown page

Finding Crawl Seeds web graph

Fraction of new pages discovered

Overhead

WebBase project, two crawls of ~1,000,000 pages, one week apart

4-level BFS crawl

Summary web graph

Two graph properties make local PageRank approximation hard

The Web Graph is not suitable for

local PR approximation

The Reverse Web graph is suitable

for local PR approximation

RPR finds nodes that

- have short paths to many other nodes
- frequently the only gateways to these nodes
Applications of RPR

Thanks! web graph

Appendix web graph

Proof – High in-degree Deterministic algorithms web graph

x1

x2

x3

xm

- By reduction from the majority-by-a-margin problem

Input:

Output: the majority

At least queries are needed

The Reduction web graph

1

1

0

m

X=

Gx=

W1

W2

Wm

V1

V2

V3

u

- A - Alg. that calculates local PR
- B - Alg. that computes majority-by-a-margin

The Reduction web graph

1

1

0

m

X=

Claim 1: Let |x| be the

number of 1’s in x. Then,

Gx=

W1

W2

Wm

V1

V2

V3

u

Claim 2: When ,

Proof Cont. web graph

- Given an input x, B simulates A on Gx, u
- If PRx(u) ≥ p1 => The majority bit of x is 1
- If PRx(u) ≤ p0 => The majority bit of x is 0
- It means that the maximum number of queries A uses ≥

Proof – Slow PR Conversion Randomized algorithms web graph

x1

x2

x3

xm

- By reduction from the OR problem

Input:

Output:

queries are needed even for

randomized algorithms

The Reduction web graph

0

1

0

m

X=

Gx=

Sm

S1

……

T

- A - Alg. that calculates local PR
- B - Alg. that computes the OR function

u

The Reduction web graph

0

1

0

m

X=

Gx=

Claim 1: Let |x| be the

number of 1’s in x. Then,

Sm

S1

……

T

Claim 2: When ,

u

Proof Cont. web graph

- Given an input x, B simulates A on Gx, u
- If PRx(u) ≥ p1 => OR=1
- If PRx(u) ≤ p0 => OR=0
- It means that the maximum number of queries A uses ≥

Proof – Slow PR Convergence Deterministic algorithms web graph

x1

x2

x3

xm

- By reduction from the majority-by-a-margin problem

Input:

Output: the majority

At least queries are needed

The Reduction web graph

1

1

0

m

X=

w1

w2

w3

w4

Gx=

wm-1

wm

……

- A - Alg. that calculates local PR
- B - Alg. that computes majority-by-a-margin

……

……

……

u

The Reduction web graph

1

1

0

m

X=

Claim 1: Let |x| be the

number of 1’s in x. Then,

w1

w2

w3

w4

wm-1

wm

……

……

……

……

Claim 2: When ,

u

Proof Cont. web graph

- Given an input x, B simulates A on Gx, u
- If PRx(u) ≥ p1 => The majority bit of x is 1
- If PRx(u) ≤ p0 => The majority bit of x is 0
- It means that the maximum number of queries A uses ≥

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